In this paper, we are concerned with elliptic equations of p-Laplace type with measure data, which is given by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-\operatorname {div}\big (a(x)(|\nabla u|^2+s^2)^{\frac{p-2}{2}}\nabla u\big )=\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msup> <mrow> <mo>div</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>s</mi> <mn>2</mn> </msup> <msup> <mrow> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$s\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Under the assumption that the modulus of continuity of the coefficient a(x) in the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-mean sense satisfies the Dini condition, we prove a new comparison estimate and use it to derive interior and global gradient pointwise estimates by Wolff potential for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and Riesz potential for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1<p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, respectively. Our interior gradient pointwise estimates can be applied to a class of singular quasilinear elliptic equations with measure data given by <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$-\operatorname {div}(A(x,\nabla u))=\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mo>div</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>. We generalize the results in the papers of Duzaar and Mingione [Amer. J. Math. 133, 1093–1149 (2011)], Dong and Zhu [J. Eur. Math. Soc. 26, 3939–3985 (2024)], and Nguyen and Phuc [Arch. Rational Mech. Anal. (2023) 247:49], etc., where the coefficient is assumed to be Dini continuous. Moreover, we establish interior and global modulus of continuity estimates of the gradients of solutions.

Gradient continuity estimates for elliptic equations of p-Laplace type with measure data

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发表日期：

2026年6月26日

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Longjuan Xu,
Yirui Zhao

摘要

In this paper, we are concerned with elliptic equations of p-Laplace type with measure data, which is given by $-\operatorname {div}\big (a(x)(|\nabla u|^2+s^2)^{\frac{p-2}{2}}\nabla u\big )=\mu $ $- {div (a (x) (| \nabla u |}^{2} + s^{2})^{\frac{p - 2}{2}} \nabla u) = μ$ with $p>1$ $p > 1$ and $s\ge 0$ $s \geq 0$ . Under the assumption that the modulus of continuity of the coefficient a(x) in the $L^2$ $L^{2}$ -mean sense satisfies the Dini condition, we prove a new comparison estimate and use it to derive interior and global gradient pointwise estimates by Wolff potential for $p\ge 2$ $p \geq 2$ and Riesz potential for $1<p<2$ $1 < p < 2$ , respectively. Our interior gradient pointwise estimates can be applied to a class of singular quasilinear elliptic equations with measure data given by $-\operatorname {div}(A(x,\nabla u))=\mu $ $- div (A (x, \nabla u)) = μ$ . We generalize the results in the papers of Duzaar and Mingione [Amer. J. Math. 133, 1093–1149 (2011)], Dong and Zhu [J. Eur. Math. Soc. 26, 3939–3985 (2024)], and Nguyen and Phuc [Arch. Rational Mech. Anal. (2023) 247:49], etc., where the coefficient is assumed to be Dini continuous. Moreover, we establish interior and global modulus of continuity estimates of the gradients of solutions.